Calculus Summer Math Packet 2017 – Introduction to Calculus
1. Simplify the following rational expression:
1
𝑥 −
1
−
𝑥2
1
3
1
9
3. Simplify the exponential expression:
3
√27𝑎6 𝑏12
5. . Factor each expression completely:
9𝑥 3 − 9𝑥 2 − 𝑥 − 1
𝑚4 − 16
7. Solve for h:
page 1
2. Rationalize the denominator:
7
3√7
4. Simplify the logarithmic expression:
log 5 27 − 2 log 5 3
6. Solve for x in each equation:
1
= 21−2𝑥
8
log 3 𝑥 + 2 log 3 2 = 4 log 3 2
8. Solve by completing the square:
𝑆 = 2𝜋𝑟 2 + 2𝜋𝑟ℎ
9. Complete the square to rewrite the equation in
vertex form:
𝑥 2 + 𝑥 − 2𝑦 = 0
𝑥 2 − 8𝑥 + 3 = 0
10. Find all solutions, real and imaginary, for:
𝑥 4 − 9𝑥 2
𝑚4 − 16
Calculus Summer Math Packet 2017 – Introduction to Calculus
11. Write an equation for a line in standard form
passing through (2,2) and perpendicular to the line:
2𝑥 − 3𝑦 = 6
13. For the circle (𝑥 + 3)2 + (𝑦 − 2)2 = 10, expand and
write in general, quadratic form.
page 2
12. Write an equation for a line in slope-intercept form
passing through (4, 0) and the midpoint of the line
segment from (−1,5) to (3,7).
14. Find the solution to each equation:
𝑥 2 + 3𝑥 = ̶ 1
3
9
=
𝑥+1
4𝑥 + 5
15. Find the solution to the following inequality:
16. Find the solution(s) for each statement:
6
>4
𝑥−2
17. Find the solution to the system of linear equations:
{
𝑥 − 3𝑦 + 10 = 0
3𝑥 − 𝑦 + 6 = 0
19. Describe the vertices and foci of the ellipse with the
following equation:
(𝑥 − 3)2 + ( 𝑦 + 2)2 = 4
|2 − 𝑥| ≥ 4
18. Shade the region that satisfies this system of
inequalities:
{
𝑥 − 3𝑦 + 10 > 0
3𝑥 − 𝑦 + 6 < 0
20. Find the domain and range for the function:
𝑔(𝑥) =
1 𝑥
𝑒
2
Calculus Summer Math Packet 2017 – Introduction to Calculus
21. Let 𝑓(𝑥) = 𝑥 2 and 𝑔(𝑥) = √1 − 𝑥
a. Describe the domain and range of each function.
b. Form composite functions 𝑓 ∘ 𝑔 and 𝑔 ∘ 𝑓.
c. Describe the domain and range for each composite
function 𝑓 ∘ 𝑔 and 𝑔 ∘ 𝑓.
23. Simplify:
𝑓( 𝑥+ℎ)−𝑓(𝑥)
ℎ
where 𝑓(𝑥) =
1
.
𝑥+1
page 3
22. Find all solutions (real and complex ) to the
equation HINT: Use Rational Root Theorem:
12𝑥 3 − 23𝑥 2 − 3𝑥 + 2 = 0
24. Sketch the graphs of each function:
a. ℎ(𝑥) = 𝑥 2 − 4𝑥 + 3
b. 𝑓(𝑥) = 2𝑥 3 + 3
c. 𝑔(𝑥) = | 4𝑥 + 6|
25. Make a sketch of each function, then write the
inverse of the function and show its graph:
1
a. 𝑔(𝑥) = (𝑥+2)2 , 𝑥 ≥ 2
26. Accurately graph: ℎ(𝑥) =
𝑥 2 −2𝑥+1
𝑥 2 −3𝑥+2
b. ℎ(𝑥) = 𝑒 𝑥
27. Use synthetic division to divide :
𝑥 5 − 4𝑥 4 + 𝑥 3 − 7𝑥 + 1
𝑥+2
28. Use long division to divide:
𝑥 5 − 𝑥 4 + 𝑥 3 + 2𝑥 2 − 𝑥 + 4
𝑥3 + 1
29. Sketch the graphs of each transformation of 𝑦 =
sin 𝑥:
30. Find the intersection(s) by graphing of the functions
when 0≤ 𝑥 ≤ 2𝜋 :
𝑥
a. Sketch of 𝑦 = sin ( )
2
b. Sketch of 𝑦 = 2 sin(𝑥) − 3
{
𝑦 = sin 𝑥
𝑦 = cos 𝑥
Calculus Summer Math Packet 2017 – Introduction to Calculus
31. Solve for x on the interval 0≤ 𝑥 ≤ 2𝜋
a. sin x =
page 4
32. Simplify the trig identities:
tan 𝑥 csc 𝑥 − cos 𝑥
√3
2
b. cos x = −
√2
2
sin 𝑥 cos 𝑥
+
csc 𝑥 sec 𝑥
33. A kite is 100 m above the ground. If there are 200 m
of string out, what is the angle between the string and
the horizontal (assume that the string is perfectly
straight)?
34. A rectangle is inscribed inside the parabola 𝑦 = 4 −
𝑥 2 , so that one of its vertices is at (𝑥, 4 − 𝑥 2 ).
a. Represent the area of the rectangle as a function of x.
b. Describe the domain for this function because the
rectangle is inscribed in the parabola.
35. Two points on a coordinate plane are located at
(5,4) and (−2, −20). Find the distance between the
points.
36. Two cars start moving from the same point. One
travels south at 100 kph, the other, west, at 50 kph. How
far apart are they in two hours?
37. The figure at the right is composed of a rectangle and a semicircle.
a. Write the formula for the perimeter of the figure.
b. Write the formula for the area of the figure.
r
r
38. A water tank has the shape of a cone. The tank is 10 m high and has a radius of 3 m at the top. If the water is
5 m deep in the middle, what is the surface area of the top of the water?
Calculus Summer Math Packet 2017 – Introduction to Calculus
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Calculus Summer Math Packet 2017 – Introduction to Calculus
ANSWER SHEET pg 2
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Calculus Summer Math Packet 2017 – Introduction to Calculus
ANSWER SHEET pg 3
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